Page 340 - Determinants and Their Applications in Mathematical Physics

A.6 The Generalized Geometric Series and Eulerian Polynomials 325

Lawden’s function S m (x) is defined as follows:
S m (x)=(1 − x) m+1 ψ m (x), m ≥ 0. (A.6.11)

It follows from (A.6.5) that S m is a polynomial of degree m in (1 − x) and
hence is also a polynomial of degree m in x. Lawden’s investigation into
the properties of ψ m and S m arose from the application of the z-transform
to the solution of linear difference equations in the theory of sampling
servomechanisms.
The Eulerian polynomial A m (x), not to be confused with the Euler
polynomial E m (x), is defined as follows:

A m (x)=(1 − x) m+1 φ m (x), m ≥ 0, (A.6.12)
A m (x)= S m (x), m > 0,
A 0 (x)=1,
S 0 (x)= x, (A.6.13)
m

A m (x)= A mn x , (A.6.14)
n
n=1
where the coefficients A mn are the Eulerian numbers which are given by
the formula
n−1
m +1

A mn = (−1) r (n − r) , m ≥ 0,n ≥ 1,
m
r
r=0
= A m,m+1−n . (A.6.15)
These numbers satisfy the recurrence relation
A mn =(m − n +1)A m−1,n−1 + nA m−1,n . (A.6.16)
The first few Eulerian polynomials are

A 1 (x)= S 1 (x)= x,
2
A 2 (x)= S 2 (x)= x + x ,
3
2
A 3 (x)= S 3 (x)= x +4x + x ,
2
4
3
A 4 (x)= S 4 (x)= x +11x +11x + x ,
4
5
3
2
A 5 (x)= S 5 (x)= x +26x +66x +26x + x .
S m satisfies the linear recurrence relation
m−1
m
m−1
(1 − x)S m =(−1) (−1) r (1 − x) m−r
r S r
r=0
and the generating function relation
∞
x(x − 1) S m (x)u m
V = = ,
x − e u(x−1) m!
m=0